Let k ≥ 2 be an integer, a k-decomposition (G1,G2, . . . ,Gk) of the complete graph Kn is a partition of its edge set to form k spanning subgraphs G1,G2, . . . ,Gk. In this contribution, we investigate the Nordhaus–Gaddum-type inequality of a k-decomposition of Kn for the general Zagreb index and a 2-decomposition for the Zagreb co-indices, respectively. The corresponding extremal graphs are ch...

Let F ⊂ 2 be a 3-wise 2-intersecting Sperner family. It is proved that |F| ≤ { ( n−2 (n−2)/2 ) if n even, ( n−2 (n−1)/2 ) + 2 if n odd holds for n ≥ n0. The unique extremal configuration is determined as well.

Communicated by the Managing Editors We investigate those graphs G, with the property that any tree on N vertices occurs as subgraph of G,. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $z log II and nl+l/log log %, respectively. A typical question in extremal graph theory1 is one wh...

We first obtain Liouville type results for stable entire solutions of the biharmonic equation −∆2u = u−p in R for p > 1 and 3 ≤ N ≤ 12. Then we consider the Navier boundary value problem for the corresponding equation and improve the known results on the regularity of the extremal solution for 3 ≤ N ≤ 12. As a consequence, in the case of p = 2, we show that the extremal solution u∗ is regular w...

A cusp form f(z) of weight k for SL2(Z) is determined uniquely by its first ` := dimSk Fourier coefficients. We derive an explicit bound on the nth coefficient of f in terms of its first ` coefficients. We use this result to study the nonnegativity of the coefficients of the unique modular form of weight k with Fourier expansion Fk,0(z) = 1 + O(q ). In particular, we show that k = 81632 is the ...

The /-connectivity ~c/(G) of a graph G of order p ~ > / I is the minimum number of vertices that are required to be deleted from G to produce a graph with at least f components or with fewer than /' vertices. Extremal graphs are investigated and bounds established on the number of edges in graphs of given order and f-connectivity. @ 1999 Elsevier Science B.V, All rights reserved